Periodic energy space on an interval

PDF p. 2: §1.2, definition of H¹p(−π,π)

Assume the Axiom of Choice The Axiom of Choice. By The Axiom of Choice implies countable choice, this implies Countable Choice, as required by the Sobolev interfaces below. Let $I=(-\pi ,\pi )$ and write $H^1(I)=W^{1,2}(I;\mathbb{C})$: its elements are complex $L^2$ equivalence classes whose first weak derivative also belongs to $L^2(I)$, equipped with the norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1(I)}=(\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(I)}^2+\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(I)}^2{)}^{1\mathrm{/}2}$ Integer-order Sobolev spaces and their norms. This expression is a well-defined norm by The Sobolev norm descends to equivalence classes, applied with $n=1$, $k=1$, $p=2$ and complex scalars. The periodic energy space is $H_p^1(-\pi ,\pi )=\{v\in H^1(I):v^{\ast }(-\pi )=v^{\ast }(\pi )\},$ with the restricted $H^1(I)$ norm. Here $v^{\ast }$ is the unique absolutely continuous representative on $[-\pi ,\pi ]$ of the class $v$; its endpoint values are the Sobolev traces The trace of a one-dimensional Sobolev function is the pair of endpoint values.

By The trace of a one-dimensional Sobolev function is the pair of endpoint values, under the stated Axiom of Choice every $v\in H^1(I)$ has a uniquely determined absolutely continuous representative on the closed interval, and the endpoint trace map is linear and bounded. Therefore the equality of its endpoint values is independent of the representative of the $L^2$ class. Define $D:H^1(I)\to \mathbb{C}$ by $Dv=v^{\ast }(-\pi )-v^{\ast }(\pi )$. Linearity of the trace gives $D(\alpha u+\beta v)=\alpha Du+\beta Dv$ for $u,v\in H^1(I)$ and $\alpha ,\beta \in \mathbb{C}$. Thus $H_p^1(-\pi ,\pi )=\ker D$ is a linear subspace by The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial; the zero class belongs to it. The norm is well defined by restriction of the Sobolev norm in Integer-order Sobolev spaces and their norms, whose norm axioms and independence of representatives are supplied by The Sobolev norm descends to equivalence classes under Countable Choice, already obtained from the assumed Axiom of Choice.